Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

• Main Authors: Colding, Tobias (Contributor), Naber, Aaron Charles (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Springer-Verlag, 2013-08-21T19:26:59Z.
• Subjects: Article
• Online Access: Get fulltext

Original manuscript January 6, 2012
Consider a limit space (M[subscript α],g[subscript α],p[subscript α]) [superscript GH over →] (Y, d[subscript Y], p), where the M[superscript n over α] have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point p ∈ Y are known to be metric cones C(X), however they need not be unique. Let [superscript dash over Ω]Y,[subscript p] ⊆ M[subscript GH] be the closed subset of compact metric spaces X which arise as cross sections for the tangents cones of Y at p. In this paper we study the properties of [superscript dash over Ω]Y,[subscript p] . In particular, we give necessary and sufficient conditions for an open smooth family Ω ≡ (X, g[subscript s]) of closed manifolds to satisfy [superscript dash over Ω] = [superscript dash over Ω]Y,[subscript p] for some limit Y and point p ∈ Y as above, where [superscript dash over Ω] is the closure of Ω in the set of metric spaces equipped with the Gromov-Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces (Y [superscript n], d Y, p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n − 2 a tangent cone at p of the form R[superscript k] × C(X[superscript n-k-1]), where X[superscript n-k-1] is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space Y based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space (Y[superscript 5], d[subscript Y], p), such that at p the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over CP[superscript 2]#[superscript dash over CP][superscript 2] while others are homeomorphic to cones over S[superscript 4].
National Science Foundation (U.S.) (Grant DMS 0606629)
National Science Foundation (U.S.) (Grant DMS 1104392)
National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774)
National Science Foundation (U.S.) (Postdoctoral Fellowship)